Math Problem Statement
Let $z$ and $w$ be complex numbers such that $|z| = |w| = 1$ and $zw \ne -1.$
(a) Prove that $\overline{z} = \frac{1}{z}$ and $\overline{w} = \frac{1}{w}.$
(b) Prove that $\frac{z + w}{zw + 1}$ is a real number.
Solution
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Conjugates
Modulus
Real and Imaginary Parts
Formulas
$|z|^2 = z\overline{z} = 1$
$\overline{z} = \frac{1}{z}$
$\overline{\left(\frac{z + w}{zw + 1}\right)} = \frac{z + w}{zw + 1}$
Theorems
Modulus of Complex Numbers
Properties of Conjugates
Real Number Condition
Suitable Grade Level
Undergraduate level